(The) polynomial interpolating at the zeros of orthogonal polynomials

Abstract

The main purpuse of this work is to introduce a new approximation method and to estimate its convergence. We show that this method is the polynomial interpolation at zeros of orthogonal polynomials. In this method, the interpolation polynomials of all continuous function on a finite closed interval converges to a give function in $L_2$-sence. Also if $lim_{\delta\rightarrow 0}\sqrt{n}\omega(f;\frac{1}{n})=0$, where ω(f;δ) is modulus of continuity, then interpolation of $f(x)$ at zeros of Jacobi orthogonal polynomial $P^{(\alpha,\beta)}_{n+1}$ with -1<α,β<0 converges to f(x).